The inverse scattering method for solving nonlinear evolution equations, which are a special kind of differential equations, goes back to Gardner et al. It can be thought of as a generalization of Fourier's method for solving the heat equation. The idea is that the time-evolution of a signal governed by some evolution equation might be trivial in the (nonlinear) Fourier domain where a space-time signal is expressed in terms of suitably chosen waves. To determine the temporal evolution of a spatial signal (or vice versa, the spatial evolution of a temporal signal) in a medium, a suitable nonlinear Fourier transform of the signal is computed and the temporal (or spatial) evolution of the signal is performed in the nonlinear Fourier domain. Eventually, an inverse nonlinear Fourier transform is used to recover the evolved signal at the desired time. The nonlinear Fourier transforms disclosed herein are true generalizations of the common Fourier transform. They enable the analysis and synthesis of functions in terms of non-sinusoidal wave forms that are fundamental in several important physical setups, such as light waves in optical fiber. It is therefore desirable to provide improved techniques for implementing nonlinear Fourier transforms and inverse nonlinear Fourier transforms.